Question

Transplant operations have become routine. One common transplant operation is for kidneys. The most dangerous aspect of the procedure is the possibility that the body may reject the new organ. There are several new drugs available for such circumstances and the earlier the drug is administered, the higher the probability of averting rejection. The New England Journal of Medicine recently reported the development of a new urine test to detect early warning signs that the body is rejecting a transplanted kidney. However, like most other tests, the new test is not perfect. In fact, 20% of negative tests and 5% of positive tests prove to be incorrect. Physicians know that in about 31% of kidney transplants the body tries to reject the organ. If the new test has a positive result (indicating early warning of rejection), what is the probability that the body is attempting to reject the kidney?

Answer 1

Answer:

Given that,

Transplant operations have become routine. One common transplant operation is for kidneys. The most dangerous aspect of the procedure is the possibility that the body may reject the new organ.

20% of negative tests and 5% of positive tests prove to be incorrect. Physicians know that in about 31% of kidney transplants the body tries to reject the organ. If the new test has a positive result (indicating early warning of rejection),.

Definition:

P(A | B) = Probability of A being true given that B is true.

Positive / Negative = test results are positive / negative.

Accepted / Rejected = the organ is accepted / rejected.

What is Given:

If the organ is actually rejected, then a correct test result would be positive and an incorrect result would be negative.

If the organ is actually accepted, then a correct test result would be negative and an incorrect result would be positive.

(1).

20 % of the rejected organs will incorrectly receive negative test results.

This means,

Probability that the test would be negative given the organ is rejected is 0.02

P(negative|rejected) = 0.2

P(positive|rejected) = 1-0.2

= 0.8

(2).

5 % of the accepted organs will incorrectly receive positive result.

P(positive|accepted) = 0.05

(3).

Physicians know that 31 % of the transplants are going to be rejected.

P(rejected) = 0.31

P(accepted) = 1-0.31

= 0.67

What is asked:

P(rejected|positive)

Solution:

Use Bayes theorem:

=0.88099

=0.881(approximately)